8.1.2 L-C Resonators

An EM field can be generated using an L-C resonator, or L-C tank. One topology for an L-C tank is a parallel R-L-C construction, as shown in Figure 50.

Figure 50. Electrical Model of the L-C Tank Sensor

A resonant oscillator can be constructed by combining a frequency selective circuit (resonator) with a gain block in a closed loop. The criteria for oscillation are: (1) loop gain > 1, and (2) closed loop phase shift of 2π radians. The R-L-C resonator provides the frequency selectivity and contributes to the phase shift. At the resonance frequency, the impedance of the reactive components (L and C) cancels, leaving only R_P, the lossy (resistive) element in the circuit. The voltage amplitude is maximized at this frequency. The R_P can be used to determine the sensor drive current for a given oscillation amplitude. A lower R_P requires a larger sensor current to maintain a constant oscillation amplitude. The sensor oscillation frequency is given by:

Equation 1.

where

• C is the sensor capacitance (C_SENSOR + C_PARASITIC)
• L is the sensor inductance
• Q is the quality factor of the resonator. Q can be calculated by:

Equation 2.

where

• R_P is the AC parallel resistance of the LC resonator at the operating frequency.

Texas Instruments' WEBENCH design tool can be used for coil design, in which the parameter values for R_P, L and C are calculated. See http://www.ti.com/webench.

R_P is a function of target distance, target material, and sensor characteristics. Figure 51 shows an example of R_P variation based on the distance between the sensor and the target. The graph represents a 14 mm diameter PCB coil (23 turns, 4 mil trace width, 4 mil spacing between traces, 1 oz. copper thickness, on FR4 material). This curve is a typical response where the target distance scales based on the sensor size and the sensor R_P scales based on the free-space of the inductor.

Figure 51. Example R_P vs. Distance with a 14 mm PCB Coil and 2 mm Thick Stainless Steel Target

It is important to configure the sensor current drive so that the sensor will still oscillate at the minimum R_P value (which typically occurs with maximum target interaction). As an example, if the closest target distance in a system with the response shown in Figure 51 is 1mm, then the sensor current drive needs to support a R_P value is 5 kΩ. Both the minimum and maximum R_P conditions should have oscillation amplitudes that are within the device operating range. See section Sensor Current Drive Control for details on setting the current drive.

The inductance that is measured by the LDC is:

Equation 3.

where

• L(d) is the measured sensor inductance, for a distance d between the sensor coil and target
• L_inf is the inductance of the sensing coil without a conductive target (target at infinite distance)
• M(d) is the mutual inductance
• ƒ_SENSOR = sensor oscillation frequency for a distance d between the sensor coil and target
• C = C_SENSOR + C_PARASITIC

Figure 52 shows an example of variation in sensor frequency and inductance as a function of distance for a 14 mm diameter PCB coil (23 turns, 4 mil trace width, 4 mil spacing between traces, 1 oz copper thickness, FR4 material). The frequency and inductance graphs will scale based on the sensor free-space characteristics, and the target distance scales based on the sensor diameter.

Figure 52. Example Sensor Frequency, Inductance vs. Target Distance
with 14 mm PCB Coil and 1.5 mm Thick Aluminum Target

The Texas Instruments Application Notes LDC Sensor Design and LDC Target Design provide more information on construction of sensors and targets charactersitics to consider based on system requirements.